ACT Math · Intermediate Algebra
Arithmetic and geometric sequences
Finding the nth term of arithmetic and geometric sequences.
What's tested in this subtopic
Arithmetic sequences add a constant; geometric sequences multiply by a constant. Items test the nth-term formulas and occasionally the sum formulas.
Tactics that actually move your score
Memorize: arithmetic nth term aₙ = a₁ + (n − 1)d; geometric nth term aₙ = a₁ · r^(n − 1). The single biggest leverage point on most subtopics isn't learning more math — it's recognizing the test's preferred surface forms quickly enough that you don't burn 30 seconds re-reading the question. The first time you see a particular phrasing it might take you a full minute. The tenth time you see it, you should be reaching for your method before you've finished the sentence. Repetition is what builds that recognition. Fifteen problems in a row of the same shape is more useful than fifty mixed.
Practice questions (14)
- Easy An arithmetic sequence has first term 8 and common difference 2. What is the 17th term?
- Easy An arithmetic sequence has first term 2 and common difference 4. What is the 12th term?
- Medium An arithmetic sequence has first term 9 and common difference 7. What is the 13th term?
- Medium An arithmetic sequence has first term 5 and common difference 7. What is the 11th term?
- Medium An arithmetic sequence has first term 5 and common difference 6. What is the 11th term?
- Hard An arithmetic sequence has first term 3 and common difference 2. What is the 19th term?
- Easy An arithmetic sequence has first term 6 and common difference 3. What is the 18th term?
- Easy An arithmetic sequence has first term 2 and common difference 8. What is the 13th term?
- Medium An arithmetic sequence has first term 9 and common difference 4. What is the 16th term?
- Medium An arithmetic sequence has first term 10 and common difference 5. What is the 18th term?
- Medium An arithmetic sequence has first term 4 and common difference 4. What is the 9th term?
- Hard An arithmetic sequence has first term 6 and common difference 2. What is the 13th term?
- Easy An arithmetic sequence has first term 10 and common difference 6. What is the 16th term?
- Easy An arithmetic sequence has first term 9 and common difference 4. What is the 8th term?
How to drill
Work through the questions above untimed. After each one, read the worked solution from start to finish — even when you got it right. Note which solution method you used, and which method we used; if they differ, ask yourself which would have been faster on test day. Speed in ACT math comes from shortening your method-selection step, not from doing arithmetic faster. Most fast students are doing the same arithmetic everyone else is — they're just spending less time deciding what to do.
Once you can clear the easy and medium items in this subtopic at 90% accuracy, attempt a timed mini-set of ten hard items at 75 seconds each. If you finish in time and score 7+ correct, you've effectively mastered the subtopic for test purposes and can move on.